Mixing
Electromagnetic electron wave
In plasma physics, an electromagnetic electron wave is a wave in a plasma which has a magnetic field component and in which primarily the electrons oscillate.
In an unmagnetized plasma, an electromagnetic electron wave is simply a light wave modified by the plasma. In a magnetized plasma, there are two modes perpendicular to the field, the O and X modes, and two modes parallel to the field, the R and L waves.
Contents
1 Cut-off frequency and critical density
2 O wave
3 X wave
4 R wave and L wave
5 Dispersion relations
6 See also
7 References
Cut-off frequency and critical density
In an unmagnetized plasma for the high frequency or low electron density limit, i.e. for ω≫ωpe=(nee2/meϵ0)1/2{displaystyle omega gg omega _{pe}=(n_{e}e^{2}/m_{e}epsilon _{0})^{1/2}}
or
ne≪meω2ϵ0/e2{displaystyle n_{e}ll m_{e}omega ^{2}epsilon _{0},/,e^{2}}
nc=εomee2ω2{displaystyle n_{c}={frac {varepsilon _{o},m_{e}}{e^{2}}},omega ^{2}}(SI units)
If the critical density is exceeded, the plasma is called over-dense.
In a magnetized plasma, except for the O wave, the cut-off relationships are more complex.
O wave
The O wave is the ordinary wave in the sense that its dispersion relation is the same as that in an unmagnetized plasma. It is plane polarized with
E1 || B0. It has a cut-off at the plasma frequency.
X wave
The X wave is the "extraordinary" wave because it has a more complicated dispersion relation. It is partly transverse (with E1⊥B0)
and partly longitudinal. As the density is increased, the phase velocity rises from c until the cut-off at ωR is reached. As the density is further increased, the wave is evanescent until the resonance at the upper hybrid frequency ωh. Then it can propagate again until the second cut-off at ωL. The cut-off frequencies are given by [2]
- ωR=12[ωc+(ωc2+4ωp2)12]ωL=12[−ωc+(ωc2+4ωp2)12]{displaystyle {begin{aligned}omega _{R}&={frac {1}{2}}left[omega _{c}+left(omega _{c}^{2}+4omega _{p}^{2}right)^{frac {1}{2}}right]\omega _{L}&={frac {1}{2}}left[-omega _{c}+left(omega _{c}^{2}+4omega _{p}^{2}right)^{frac {1}{2}}right]end{aligned}}}
![{displaystyle {begin{aligned}omega _{R}&={frac {1}{2}}left[omega _{c}+left(omega _{c}^{2}+4omega _{p}^{2}right)^{frac {1}{2}}right]\omega _{L}&={frac {1}{2}}left[-omega _{c}+left(omega _{c}^{2}+4omega _{p}^{2}right)^{frac {1}{2}}right]end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e3c7d4f2e54dcfba17e48a86c9574d298c1e2252)
where ωc{displaystyle omega _{c}}
R wave and L wave
The R wave and the L wave are right-hand and left-hand circularly polarized, respectively. The R wave has a cut-off at ωR (hence the designation of this frequency) and a resonance at ωc. The L wave has a cut-off at ωL and no resonance. R waves at frequencies below ωc/2 are also known as whistler modes. [3]
Dispersion relations
The dispersion relation can be written as an expression for the frequency (squared), but it is also common to write it as an expression for the index of refraction ck/ω (squared).
| Conditions | Dispersion relation | Name |
|---|---|---|
B→0=0{displaystyle {vec {B}}_{0}=0} |
ω2=ωp2+k2c2{displaystyle omega ^{2}=omega _{p}^{2}+k^{2}c^{2}} |
Light wave |
k→⊥B→0, E→1‖B→0{displaystyle {vec {k}}perp {vec {B}}_{0}, {vec {E}}_{1}|{vec {B}}_{0}} |
c2k2ω2=1−ωp2ω2{displaystyle {frac {c^{2}k^{2}}{omega ^{2}}}=1-{frac {omega _{p}^{2}}{omega ^{2}}}} |
O wave |
k→⊥B→0, E→1⊥B→0{displaystyle {vec {k}}perp {vec {B}}_{0}, {vec {E}}_{1}perp {vec {B}}_{0}} |
c2k2ω2=1−ωp2ω2ω2−ωp2ω2−ωh2{displaystyle {frac {c^{2}k^{2}}{omega ^{2}}}=1-{frac {omega _{p}^{2}}{omega ^{2}}},{frac {omega ^{2}-omega _{p}^{2}}{omega ^{2}-omega _{h}^{2}}}} |
X wave |
k→‖B→0{displaystyle {vec {k}}|{vec {B}}_{0}}  (right circ. pol.) |
c2k2ω2=1−ωp2/ω21−ωc/ω{displaystyle {frac {c^{2}k^{2}}{omega ^{2}}}=1-{frac {omega _{p}^{2}/omega ^{2}}{1-omega _{c}/omega }}} |
R wave (whistler mode) |
k→‖B→0{displaystyle {vec {k}}|{vec {B}}_{0}} (left circ. pol.) |
c2k2ω2=1−ωp2/ω21+ωc/ω{displaystyle {frac {c^{2}k^{2}}{omega ^{2}}}=1-{frac {omega _{p}^{2}/omega ^{2}}{1+omega _{c}/omega }}} |
L wave |
See also
- Appleton-Hartree equation
- List of plasma (physics) articles
References
^ Chen, Francis (1984). Introduction to Plasma Physics and Controlled Fusion, Volume 1 (2nd ed.). Plenum Publishing Corporation. p. 116. ISBN 0-306-41332-9..mw-parser-output cite.citation{font-style:inherit}.mw-parser-output .citation q{quotes:"""""""'""'"}.mw-parser-output .citation .cs1-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .citation .cs1-lock-limited a,.mw-parser-output .citation .cs1-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .citation .cs1-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration{color:#555}.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration span{border-bottom:1px dotted;cursor:help}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/12px-Wikisource-logo.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output code.cs1-code{color:inherit;background:inherit;border:inherit;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;font-size:100%}.mw-parser-output .cs1-visible-error{font-size:100%}.mw-parser-output .cs1-maint{display:none;color:#33aa33;margin-left:0.3em}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-right{padding-right:0.2em}
^ Chen, Francis (1984). Introduction to Plasma Physics and Controlled Fusion, Volume 1 (2nd ed.). Plenum Publishing Corporation. p. 127. ISBN 0-306-41332-9.
^ Chen, Francis (1984). Introduction to Plasma Physics and Controlled Fusion, Volume 1 (2nd ed.). Plenum Publishing Corporation. p. 131. ISBN 0-306-41332-9.
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